Deep Learning Topological Invariants of Band Insulators

TL;DR

This paper demonstrates that deep neural networks can accurately predict topological invariants of band insulators from Hamiltonian data, capturing the underlying mathematical structures and generalizing beyond training data.

Contribution

It introduces a neural network approach to predict topological invariants in insulators, revealing that networks learn the mathematical formulas of these invariants.

Findings

Neural networks achieve over 90% accuracy in predicting invariants.

Intermediate network layers resemble mathematical formulas of topological invariants.

Models generalize well to Hamiltonians beyond the training set.

Abstract

In this work we design and train deep neural networks to predict topological invariants for one-dimensional four-band insulators in AIII class whose topological invariant is the winding number, and two-dimensional two-band insulators in A class whose topological invariant is the Chern number. Given Hamiltonians in the momentum space as the input, neural networks can predict topological invariants for both classes with accuracy close to or higher than 90%, even for Hamiltonians whose invariants are beyond the training data set. Despite the complexity of the neural network, we find that the output of certain intermediate hidden layers resembles either the winding angle for models in AIII class or the solid angle (Berry curvature) for models in A class, indicating that neural networks essentially capture the mathematical formula of topological invariants. Our work demonstrates the ability of neural networks to predict topological invariants for complicated models with local Hamiltonians as the only input, and offers an example that even a deep neural network is understandable.